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The consensus reaching process (CRP) is often utilized to mitigate conflicts among decision makers and help them reach an agreement. Almost all existing CRP models focused on obtaining a consensual complete ranking of alternatives. However, in many situations, it may be sufficient to classify alternatives into a number of ordinal classes rather than a complete ranking. To this end, this study proposes an ordinal classification-based CRP (OCCRP) with the aim to generate consensual ordinal classes of alternatives. Further, this study investigates the OCCRP in the large-scale multi-attribute group decision making context, in which flexible linguistic expression is used to model uncertain opinions of decision makers. In the OCCRP, an approach is proposed to manage flexible linguistic expression by transforming it into linguistic distribution assessment. Then, an opinion clustering and aggregation approach is used to deal with the assessment information of the large group. Next, an approach-based technique for order performance by similarity to ideal solution is used to obtain the ordinal classification of alternatives. Following this, a feedback adjustment mechanism is presented to help decision makers improve their consensus level on the ordinal classification of alternatives. Finally, a numerical example about grape wine grading problem and a comparative analysis are presented to demonstrate the validity of the proposal. [ABSTRACT FROM AUTHOR]

Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$. In this paper, we introduce and characterize $\mathcal {F}$ -(almost) greedy bases. Given such a family $\mathcal {F}$ , a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$ -greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$ , $m \in \mathbb {N}$ , and $G_m(x)$ , we have $$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$ Here, $G_m(x)$ is a greedy sum of x of order m , and $\mathbb {K}$ is the scalar field. From the definition, any $\mathcal {F}$ -greedy basis is quasi-greedy, and so the notion of being $\mathcal {F}$ -greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal {F}$ -greedy bases as being $\mathcal {F}$ -unconditional, $\mathcal {F}$ -disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal {F}$ -almost greedy bases. Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$ -greedy. For a countable ordinal $\alpha $ , we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$ , where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $. We show that for each $\alpha $ , there is a basis that is $\mathcal {S}_{\alpha }$ -greedy but is not $\mathcal {S}_{\alpha +1}$ -greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha , $$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$ [ABSTRACT FROM AUTHOR]

This article contains the description, translation, and discussion of Boston MFA 38-2-5, a limestone block found during excavation works at mastaba G(iza) 1203 carried out in 1938 under the supervision of G. A. Reisner. The object bears a dated hieratic inscription in red paint that for the first time is comprehensively translated and analysed. Using up-to-date digital image-enhancement processes, the reading of the date is improved, and certain lexical and grammatical features (the apparent use of a cardinal number instead of an ordinal, names of workforce units, etc.) are discussed. It is argued that the object had been deposited intentionally within the filling of the mastaba during its construction. The article contains a small palaeographic overview of the hieratic signs employed in the inscription. [ABSTRACT FROM AUTHOR]

Relative pairwise comparisons represent the cornerstone for several decision-making methods. Such approaches aim to support complex decision-making situations with multiple alternatives and are essential in order to provide an overall absolute evaluation of the alternatives despite the presence of experts and/or decision-makers with conflicting opinions. Moreover, when decision-maker's opinions are affected by uncertainty, there is the need to analyse the effect of such uncertain measures on the result of the decision-making process. We propose an approach based on a multi-objective optimisation problem, able to identify the presence of rank reversal issues in order to evaluate the stability of the final outcome of the decision-making process and a metric able to support experts in evaluating the effects of their uncertainty. We characterise the robustness of the ranking with respect to rank reversal by identifying a perturbation that is as small as possible, while causing the maximum number of ordinal swaps. [ABSTRACT FROM AUTHOR]

Compositional modeling of the reservoir isn’t complete without calculations of phase equilibrium and is a complex process involving many calculations. In nature, hydrocarbons don’t occur as separate components, they’re mainly mixtures. When modeling the reservoir composition for phase equilibrium calculations, in order to reduce computational costs, in practice, hydrocarbon mixtures are grouped into pseudo-components. The number of grouped pseudo-components varies from 4 to 10. This grouping process is called lumping. However, when crude oil comes to the surface, it’s important to know its detailed composition, since mixtures grouped into pseudo-components don’t allow you to know this. In this regard, modeling of the detailed composition of hydrocarbons is the main tool for understanding the detailed phase separation and design of surface facilities. In practice, this process is called delumping. In the case of this process, the detailed composition of the fluid is presented and the amount reaches up to 36 components, sometimes more. The delumping process, due to the precise separation of heavy plus fractions into carbons with single ordinal numbers (C7, C8, etc.), makes it possible to clearly recognize non-zero BIP’s in the equation of state. The new analytical approach has not previously been applied to the oils of the Caspian region, so this approach has an important role for the oil industry in the Caspian region. Analytical delumping in this paper was done to improve the PVT modeling. This paper presents the results of PVT research of Caspian oil and presents a comparison of analytical and numerical methods of delumping. As a result of the study, it was found that the analytical approach is in excellent agreement with experimental data and data from software such as PVTsim [ABSTRACT FROM AUTHOR]

A symbolic analysis of Archimedes's periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci's Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics. [ABSTRACT FROM AUTHOR]

In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated Noetherian dimension of an R-module, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension and Noetherian dimension. They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules. It is proved that a quotient finite dimensional module M has uncountably generated Krull dimension if and only if it has Krull dimension, but the values of these dimensions might differ. Similarly, a quotient finite dimensional module M has uncountably generated Noetherian dimension if and only if it has Noetherian dimension. We also show that the Noetherian dimension of a quotient finite dimensional module M with uncountably generated Noetherian dimension β is less than or equal to ω¹ + β, where ω1 is the first uncountable ordinal number. [ABSTRACT FROM AUTHOR]

Elevation of the neutrophil count is detrimental to the outcome of patients with stroke. The effect of poststroke neutrophil count on the outcome of patients with large-artery atherosclerosis (LAA) stroke is unclear. This study aims to explore the relationship of poststroke neutrophil count with the functional outcome of patients with LAA stroke, and the relationship of poststroke neutrophil count and craniocervical atherosclerotic stenosis (AS) number in these patients. The AS was defined as ≥ 50% stenosis or occlusion attributed to atherosclerosis on craniocervical large arteries. A total of 297 participants were enrolled in the cohort. In multivariable analyses, neutrophil count [adjusted relative risk (aRR) = 1.23, 95% confidence interval (CI) 1.09–1.40, p = 0.001] was an independent predictor of 90-day poor functional outcome [modified Rankin Scale (mRS) > 2 points]. The neutrophil count was significantly associated with the craniocervical AS number in a multivariable ordinal logistic regression analysis [adjusted odds ratio (aOR) = 1.41, 95% CI 1.16–1.72, p = 0.001]. The poststroke neutrophil count is a valuable predictor of 90-day poor functional outcome of patients with LAA stroke. The poststroke neutrophil count is positively correlated with the craniocervical AS number in these patients. [ABSTRACT FROM AUTHOR]

This study aims to assess the safety effect of the paved shoulder width of a 175-km six-lane rural highway in India. First, the applicability of the matched case-control (C-C) method in the case of highway segments with multiple crashes was assessed. Second, the safety effectiveness of the paved shoulder and other risk factors were evaluated using the matched C-C method. Three scenarios with different case definitions were developed using the matched C-C method and ordinal logistic regression. The study used 100-m segment-specific speed, volume, geometrics, roadside characteristics, and fatal crash data between 2016 and 2018. Developed scenario results suggested that the matched C-C method can assess crash risks on segments with multiple crashes by defining cases separately or considering the number of crashes as an ordinal variable. Results also suggested that the probability of observing crashes increases as paved shoulder width increases. However, the probability of observing multiple fatal crashes decreases for wide paved shoulder width increases. The safety effectiveness of paved shoulder width assessment showed that a wide paved shoulder had a higher risk of a fatal crash than a paved shoulder width of 1.0 to 1.5 m. The segments with unpaved shoulders were unsafe compared to segments without them. These results can be used to prepare guidelines by road-owning and construction agencies to adopt safe paved shoulder widths for six-lane and multilane highways in India. Results may also support the development of a new highway design policy concerning preferable safe paved shoulder width for multilane greenfield highways in India. [ABSTRACT FROM AUTHOR]

In previous research on two-sided matching (TSM) decision, agents' preferences were often given in the form of exact values of ordinal numbers and linguistic phrase term sets. Nowdays, the matching agent cannot perform the exact evaluation in the TSM situations due to the great fuzziness of human thought and the complexity of reality. Probability hesitant fuzzy sets, however, have grown in popularity due to their advantages in communicating complex information. Therefore, this paper develops a TSM decision-making approach with multi-attribute probability hesitant fuzzy sets and unknown attribute weight information. The agent attribute weight vector should be obtained by using the maximum deviation method and Hamming distance. The probabilistic hesitancy fuzzy information matrix of each agent is then arranged to determine the comprehensive evaluation of two matching agent sets. The agent satisfaction degree is calculated using the technique for order preference by similarity to ideal solution (TOPSIS). Additionally, the multi-object programming technique is used to establish a TSM method with the objective of maximizing the agent satisfaction of two-sided agents, and the matching schemes are then established by solving the built model. The study concludes by providing a real-world supply-demand scenario to illustrate the effectiveness of the proposed method. The proposed method is more flexible than prior research since it expresses evaluation information using probability hesitating fuzzy sets and can be used in scenarios when attribute weight information is unclear. [ABSTRACT FROM AUTHOR]

Rationale and Objectives: Multi-reader multi-case (MRMC) studies are widely used to compare the diagnostic accuracies of different imaging modalities. Although Dorfman–Berbaum–Metz (DBM) method is the most popular one among the MRMC methods, the adaptions of ANOVA statistic for linear mixed model (LMM) are not based on solid theory and the assumption of ANOVA that all groups have the same number of samples might not be met in some situations. The purpose of the article is to investigate whether the statistics for testing fixed effect in linear mixed model can yield a closer type I error rate to nominal level. Materials and Methods: We proposed to use the statistics such as likelihood ratio test (LRT) and Wald statistic to test the hypothesis of equivalence in several imaging modalities. Extensive simulations were conducted and the application to a clinical example dataset was illustrated. Results: The simulation results showed the type I error rates of Wald statistic were closer to the nominal level under many simulated situations, especially when the simulated data was ordinal and the number of diseased and non-diseased were 100. Conclusion: The Wald statistic whose degrees of freedom ware approximated by Satterthwaite's method showed competitive performance, indicating the potential of the statistic applied in DBM model for MRMC analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study modules with chain condition on uncountably generated submodules. We show that if an R -module M satisfies the ascending chain condition on uncountably generated submodules, then its Goldie dimension is less than or equal to c , where c is the first uncountable cardinal number. We also show that if a quotient finite dimensional module M satisfies the ascending chain condition on uncountably generated submodules, then it has Noetherian dimension and its Noetherian dimension is less than or equal to ω 1 , where ω 1 is the first uncountable ordinal number. We also investigate that if a quotient finite dimensional module M satisfies the descending chain condition on uncountably generated submodules, then M has Krull dimension and its Krull dimension may be any ordinal number α. [ABSTRACT FROM AUTHOR]

Gaussian mixture models (GMMs) are a popular and versatile tool for exploring heterogeneity in multivariate continuous data. Arguably the most popular way to estimate GMMs is via the expectation–maximization (EM) algorithm combined with model selection using the Bayesian information criterion (BIC). If the GMM is correctly specified, this estimation procedure has been demonstrated to have high recovery performance. However, in many situations, the data are not continuous but ordinal, for example when assessing symptom severity in medical data or modeling the responses in a survey. For such situations, it is unknown how well the EM algorithm and the BIC perform in GMM recovery. In the present paper, we investigate this question by simulating data from various GMMs, thresholding them in ordinal categories and evaluating recovery performance. We show that the number of components can be estimated reliably if the number of ordinal categories and the number of variables is high enough. However, the estimates of the parameters of the component models are biased independent of sample size. Finally, we discuss alternative modeling approaches which might be adopted for the situations in which estimating a GMM is not acceptable. [ABSTRACT FROM AUTHOR]

We study the Π31 -soundness spectra of theories. Given a recursively enumerable extension T of ACA0 , O31(T) is defined as the set of all 2-ptykes on which T is correct about well foundedness. This is a measure of how close T is to being Π31 -sound. We carry out a proof-theoretic classification of theories according to O31(T) , as well as a characterization of the sets of the form O31(T)∩Δ10. Many of the results generalize to n greater than 3. This article is part of the theme issue 'Modern perspectives in Proof Theory'. [ABSTRACT FROM AUTHOR]

This paper studies the problem of scheduling n jobs on a single machine to minimize total completion time and maximum cost, simultaneously. Each job is associated with a positional deadline that indicates the largest ordinal number of this job in any feasible schedule. The jobs have agreeable release and processing times, meaning that jobs with larger release times also have larger processing times. The agreeability assumption is reasonable since both the single-criterion problems (without positional deadline constraints) of minimizing total completion time and maximum lateness on a single machine with arbitrary release and processing times are strongly NP-hard. An O (n 3) -time Pareto optimal algorithm is presented. The previously known algorithms only solve two special cases of the agreeability assumption: either the case of equal release times in O (n 4) time, or the case of equal processing times (without positional deadline constraints) in O (n 3) time. [ABSTRACT FROM AUTHOR]

Universal Causality is a mathematical framework based on higher-order category theory, which generalizes previous approaches based on directed graphs and regular categories. We present a hierarchical framework called UCLA (Universal Causality Layered Architecture), where at the top-most level, causal interventions are modeled as a higher-order category over simplicial sets and objects. Simplicial sets are contravariant functors from the category of ordinal numbers Δ into sets, and whose morphisms are order-preserving injections and surjections over finite ordered sets. Non-random interventions on causal structures are modeled as face operators that map n-simplices into lower-level simplices. At the second layer, causal models are defined as a category, for example defining the schema of a relational causal model or a symmetric monoidal category representation of DAG models. The third layer corresponds to the data layer in causal inference, where each causal object is mapped functorially into a set of instances using the category of sets and functions between sets. The fourth homotopy layer defines ways of abstractly characterizing causal models in terms of homotopy colimits, defined in terms of the nerve of a category, a functor that converts a causal (category) model into a simplicial object. Each functor between layers is characterized by a universal arrow, which define universal elements and representations through the Yoneda Lemma, and induces a Grothendieck category of elements that enables combining formal causal models with data instances, and is related to the notion of ground graphs in relational causal models. Causal inference between layers is defined as a lifting problem, a commutative diagram whose objects are categories, and whose morphisms are functors that are characterized as different types of fibrations. We illustrate UCLA using a variety of representations, including causal relational models, symmetric monoidal categorical variants of DAG models, and non-graphical representations, such as integer-valued multisets and separoids, and measure-theoretic and topological models. [ABSTRACT FROM AUTHOR]

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We develop the abstract framework for a proof‐theoretic analysis of theories with scope beyond the ordinal numbers, resulting in an analog of ordinal analysis aimed at the study of theorems of complexity Π21$\Pi ^1_2$. This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors on the category of linear orders. Generalizing the notion of a proof‐theoretic ordinal, we define the functorial Π21$\Pi ^1_2$ norm of a theory and prove its existence and uniqueness for Π21$\Pi ^1_2$‐sound theories. From this, we further abstract a definition of the Σ21$\Sigma ^1_2$‐ andΠ21$\Pi ^1_2$‐soundness ordinals of a theory; these quantify, respectively, the maximum strength of true Σ21$\Sigma ^1_2$ theorems and minimum strength of false Π21$\Pi ^1_2$ theorems of a given theory. We study these ordinals, developing a proof‐theoretic classification theory for recursively enumerable extensions of ACA0${\mathsf {ACA_0}}$. Using techniques from infinitary and categorical proof theory, generalized recursion theory, constructibility, and forcing, we prove that an admissible ordinal is the Π21$\Pi ^1_2$‐soundness ordinal of some recursively enumerable extension of ACA0${\mathsf {ACA_0}}$ if and only if it is not parameter‐free Σ11$\Sigma ^1_1$‐reflecting. We show that the Σ21$\Sigma ^1_2$‐soundness ordinal of ACA0${\mathsf {ACA_0}}$ is ω1CK$\omega _1^{CK}$ and characterize the Σ21$\Sigma ^1_2$‐soundness ordinals of recursively enumerable, Σ21$\Sigma ^1_2$‐sound extensions of Π11$\Pi ^1_1$‐CA0${\mathsf {CA_0}}$. [ABSTRACT FROM AUTHOR]

Fuzzy‐trace theory predicts that decisionmakers process numerical information about risk at multiple levels in parallel: the simplest level, nominal (categorical some‐none) gist, and at more fine‐grained levels, involving relative comparison (ordinal less‐more gist) and exact quantities (verbatim representations). However, little is known about how individual differences in these numerical representations relate to judgments and decisions, especially involving health tradeoffs and relative risks. To investigate these differences, we administered measures of categorical and ordinal gist representations of number, objective numeracy, and intelligence in two studies (Ns = 978 and 956). In both studies, categorical and ordinal gist representations of number predicted risk judgments and decisions beyond objective numeracy and intelligence. Participants with higher scores in categorical gist were more likely to choose options to avoid cancer recurrence risks; those who were higher in ordinal gist of numbers were more likely to discriminate relative risk of skin cancer; and those with higher scores in objective numeracy were more likely to choose options that were numerically superior overall in terms of relative risk of skin cancer and of genetic risks of breast cancer (e.g., lower numerical probability of cancer). Results support parallel‐processing models that assume multiple representations of numerical information about risk, which vary in precision, and illustrate how individual differences in numerical representations are relevant to tradeoffs and risk comparisons in health decisions. These representations cannot be reduced to one another and explain psychological variations in risk processing that go beyond low versus high levels of objective numeracy. [ABSTRACT FROM AUTHOR]

The temporal and spatial distribution of the boundary temperature of chimonophilous/ thermophilic crops affect the crop phenology and planting area. Based on the surface meteorological observation data of 585 meteorological stations in Chinese from 1961 to 2020, taking the ≥0c and ≥10c thermal time, the initial and terminate days of thermal time as research indices of agricultural heat resources, based on ANUSPLIN meteorological interpolation software, the spatio-temporal evolution of boundary temperature of chimonophilous/ thermophilic crops in China were analyzed from interannual and interdecadal time scales. The influence of heat resource change on agricultural production pattern in Chinese Mainland was also discussed. The results showed that: affected by latitude and topography, the heat resources changed greatly in China. The thermal time ≥0c and ≥10c in the eastern region presented a ladder distribution with latitude, while in the western region, it was mainly affected by altitude, which decreased gradually from south to north. The thermal time ≥0c and ≥10c ranged of 700.0c8960.0c·d and 46.3c8960.0c·d respectively, with the average climate trend of 72.8c·d·10y-1 and 73.7c·d·10y-1. Within 60 years, the initial ordinal number of 0c and 10c was advanced by 9.6d and 8.4d respectively, and the terminate ordinal number was postponed by 4.8d and 7.8d, respectively. The advance of the initial day and the delay of the final day increased the number of continuous days with daily average temperature ≥0c and ≥10c by 14.4d and 16.2d in 60 years, respectively. 2011-2020cP6 periodcwas the warmest 10 years in the study period, that of the thermal time ≥0c and ≥10c increased by 6.3% and 8.2% respectively compared with P1 period (1961c1970). In P6 period, the area with thermal time ≥0c of 6000c7000c·d increased by 2.11c105km2, the area with thermal time ≥10c of 5000-6000c#183;d increased by 3.37c105km2. Under the background of climate warming, the increase of accumulated temperature has had an important impact on China's agricultural system. The layout of agricultural production and planting system should be adjusted in time to fully adapt to climate change. [ABSTRACT FROM AUTHOR]

The Spatial-Numerical Association of Response Codes (SNARC) effect - i.e., faster responses to small numbers with the left compared to the right side and to large numbers with the right compared to the left side - suggests that numbers are associated with space. However, it remains unclear whether the SNARC effect evolves from a number's magnitude or the ordinal position of a number in working memory. One problem is that, in different paradigms, the task demands influence the role of ordinality and magnitude. While single-task setups in which participants judge the parity of a displayed number indicate the importance of magnitude for the SNARC effect, evidence for ordinal influences usually comes from experiments where ordinal sequences have to be memorized or setups in which participants possess pre-existing knowledge of the ordinality of stimuli. Therefore, in this preregistered study, we employed a SNARC task without secondary ordinal sequence memorization. We dissociate ordinal and magnitude accounts by carefully manipulating experimental stimulus sets. The results indicate that even though the magnitude model better accounts for the observed data, the ordinal position seems to matter as well. Hence, numbers are associated with space in both a magnitude- and an order-respective manner, yielding a mixture of both compatibility effects. Moreover, a multiple coding framework may most accurately explain the roots of the SNARC effect. [ABSTRACT FROM AUTHOR]

Graph convolutional network is a recently developed artificial neural network method commonly used in recommendation system research. This paper points out three shortcomings of existing recommendation systems based on the graph convolutional network. 1. Existing models that take the one-hot encoding based on node ordinal numbers in the graph or encoding based on original entity attributes as input may not fully utilize the information carried by the attribute interactions. 2. Previous models update the node embeddings only by the first-order neighbors in the graph convolution layer, which is easily affected by noise. 3. Existing models do not take into account differences in user opinions. We propose an improved graph convolutional network-based collaborative filtering model to address these drawbacks. We identify inner and cross interaction between user attributes and item attributes, and then we take the vector representations of aggregated attributes graph as input. In the convolutional layer, we aggregate the second-order collaborative signals and incorporate the different user opinions. The experiments on three public datasets show that our model outperforms state-of-the-art models. [ABSTRACT FROM AUTHOR]

In order to improve the stability of the evaluation results and the gross error resistance of the algorithm in view of the widespread gross errors in geometric error evaluation, an improved self-born weighted least square method (ISWLS) is proposed in this paper. First, the nonlinear cylindrical axial model is linearized to establish the error equation of the observed values. We use the conditional equations of the independent observations found as valid information to derive the weights of the observations. The weights of the observations are subjected to least-square iteration to calculate the error values and equation parameters. Meanwhile, the ordinal numbers of the independent sets of equations in the observed equations are updated several times. By updating the ordinal number information of the conditional equations, the influence of gross error data on the solution of the equations is minimized. Through a series of experiments, the algorithm is proved to have a strong resistance to gross differences, and operation time is shorter. According to the evaluation results of cylindricity error, the uncertainty of cylindricity error was calculated by the Guide to the expression of uncertainty in measurement method (GUM)and the Monte Carlo method (MCM). Experiments show that the uncertainty results of the MCM method can verify the results assessed by the GUM method, which proves that the results of the ISWLS method are effective and robust. [ABSTRACT FROM AUTHOR]

Iron present in the drinking groundwater is attributed to the low burden of iron deficiency (ID) in Bangladesh. The supplemental anemia prevention strategies involving iron need a cautious approach due to the excess load of iron and the side effects. The present pilot study examined the potential of the guava leaf extracts to use as a natural reagent for the assessment of iron in groundwater. Eighteen households with the drinking source of groundwater were randomly selected. Guava leaves were crushed and the shreds of the leaves were mixed with the water sample. Changes of water color were photographed. Five groups were identified - 'whitish', 'shades of pink', 'shades of purple', 'light black' and the 'frank black'. The iron concentration was measured by a test kit device. Each color group was assigned a number on the ordinal scale 1-5. Statistical correlation and agreements were performed between the methods. The positive correlation (Kendall's tau b: 0.89, p,0.000) and the agreements (Kappa: 0.77, p,0.000; rho_c: 0.73, p,0.000) were observed. Guava leaf extracts may standardize an indigenous tool for a semi-quantitative measurement of groundwater iron content. Validation of the tool thus may aid in the design and evaluation of the iron supplementation and fortification programs. [ABSTRACT FROM AUTHOR]

• Sensory evaluation was performed by combining paired comparison and ranking tests. • Visual judgment about fruit disorders was studied using cherry tomato images. • Kendall's coefficient was 0.84, indicating the agreement of ranking among 12 panelists. • Ranking was influenced more by the wound area ratio than by the actual wound size. • The ranking of cherry tomato quality is predicted by the ranking prediction model. Sensory evaluation is used to identify how humans perceive and respond to stimuli. However, current judgment criteria are limited to the number of ordinal categories. These ordinal categories do not account for variations within them, making it hard to grasp the panelist's detailed preferences fully. This study aimed to develop a hierarchical ranking test to reveal human response. The quality of 100 cherry tomato images was ranked using a joint paired comparison and ranking test. One hundred images of cherry tomatoes with visible damage were investigated for sensory evaluation. Twelve untrained panelists evaluated the quality of the cherry tomato images, ranking them from 1st to 100th. As a result, the Kendall rank correlation coefficient, which evaluates the agreement of ranking among panelists, was 0.84, suggesting that the evaluation of the quality of cherry tomatoes generally showed the same trend among 12 panelists. For all 12 panelists, the proportion of the wounded area on the tomato had a more significant impact on the quality ranking than the wound area size. In addition, the quality ranking of cherry tomatoes was predicted by fine-tuning the convolutional neural network model VGG-19 with a moderately high coefficient of determination of 0.82. Our new hierarchical ranking test would facilitate detailed human evaluations, upon which ranking predictions would be made. [ABSTRACT FROM AUTHOR]

• We examine new aspects of children's ordinal understanding of a number line. • We measure if children place numbers correctly relative to each other and benchmarks. • Our new measures may reflect unique aspects of children's ordinal understanding. • Children with fewer rather than more ordinal errors scored higher on math ability. • Children with fewer vs. more ordinal errors made larger estimation gains (lower PAE). Children's performance on the number line estimation task, often measured by the percentage of absolute error, predicts their later mathematics achievement. This task may also reveal (a) children's ordinal understanding of the target numbers in relation to each other and the benchmarks (e.g., endpoints, midpoint) and (b) the ordinal skills that are a necessary precursor to children's ability to understand the interval nature of a number line as measured by percentage of absolute error. Using data from 104 U.S. kindergartners, we measured whether children's estimates were correctly sequenced across trials and correctly positioned relative to given benchmarks within trials at two time points. For both time points, we found that each ordinal error measure revealed a distinct pattern of data distribution, providing opportunities to tap into different aspects of children's ordinal understanding. Furthermore, children who made fewer ordinal errors scored higher on the Test of Early Mathematics Ability and showed greater improvement on their interval understanding of numbers as reflected by a larger reduction of percentage of absolute error from Time 1 to Time 2. The findings suggest that our number line measures reveal individual differences in children's ordinal understanding of numbers, and that such understanding may be a precursor to their interval understanding and later mathematics performance. [ABSTRACT FROM AUTHOR]

Conway's real closed field No of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems of identifying significant classes of functions that can be so extended and of defining integration for them have proven to be formidable. In this paper we address this and related unresolved issues by showing that extensions to No , and thereby integrals, exist for most functions arising in practical applications. In particular, we show they exist for a large subclass of the resurgent functions , a subclass that contains the functions that at ∞ are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as solutions to nonresonant linear and nonlinear meromorphic systems of ODEs or of difference equations. By suitable changes of variables we deal with arbitrarily located singular points. We further establish a sufficient condition for the theory to carry over to ordered exponential subfields of No more generally and illustrate the result with structures familiar from the surreal literature. The extensions of functions and integrals that concern us are constructive in nature, which permits us to work in NBG less the Axiom of Choice (for both sets and proper classes). Following the completion of the positive portion of the paper, it is shown that the existence of such constructive extensions and integrals of substantially more general types of functions (e.g. smooth functions) is obstructed by considerations from the foundations of mathematics. [ABSTRACT FROM AUTHOR]

Relational language is thought to influence mathematical skills. This study examines the association between relational language and number relation skills—knowledge of cardinal, ordinal, and spatial principles—among 104 U.S. kindergartners (5.9 years; 44% boys; 37% White, 25% Black, 14% Asian, 24% other) in the 2017–2018 academic year. Controlling for general verbal knowledge, executive function, and counting and number identification skills, relational language predicted later number relation skills, specifically number line estimation, β =.30. Relational language did not differentially predict number line estimation performance in children with low or high number relation skills, likely due to the restricted ranges of data within subgroups. Number relation skills, specifically number line estimation and number ordering, may be a pathway between relational language and mathematical skills. [ABSTRACT FROM AUTHOR]

Degraded peatland is caused by forest clearing and the construction of artificial water networks. When water management is not implemented across land uses in the entire peatland landscape, then it will be a big issue that causes a water deficit and leads to increasing droughts and fires. Effective restoration must first identify the part of Peatland Hydrological system Units (PHUs) with insufficient water storage and resources. This study used intercorrelated factors of water balance, deficit months, NDMI-NDVI indices, dry periods, recurrent fires, peat depth, and water loss conditions, as the evaluation parameters, within individual sub-PHUs to determine the most degraded areas that require intervention and restoration. Sub-PHU was determined based on the peat hydrological unity concept by identifying streamline, outlet channels, peat-depth, slopes, and network connectivity. Global hydrological data using TerraClimate and CHIRPS, combined with field observations, were used to validate and calculate each sub-PHU's water balance and dry periods. Soil moisture (NDMI), vegetation density (NDVI), and fire frequency were extracted from multispectral satellite images (e.g., Landsat 8, MODIS-Terra, and MODIS-Aqua). Each parameter was ranked by the score for each sub-PHU. The parameters that can be ranked are only the ordinal type of number. The lowest ranks indicated the most degraded sub-PHUs requiring peat rewetting interventions. [ABSTRACT FROM AUTHOR]

Multidimensional scaling (MDS) is to recover a set of points by making use of noised pairwise Euclidean distances. In some situations, the observed Euclidean distances may contain large errors or even missing values. In such cases, the order of the distances is far more important than their magnitude. Non-metric multidimensional scaling (NMDS) is then to deal with this problem by taking use of the ordinal information. The challenge of NMDS is to tackle the large number of ordinal constraints on distances (for n points, this will be of O (n 4)), which will slow down existing numerical algorithms. In this paper, we propose an ordinal weighted Euclidean distance matrix model for NMDS. By designing an ordinal weighted matrix, we get rid of the large number of ordinal constraints and tackle the ordinal constraints in a soft way. We then apply our model to image ranking. The key insight is to view the image ranking problem as NMDS in the kernel space. We conduct extensive numerical test on two state-of-the-art datasets: FG-NET aging dataset and MSRA-MM dataset. The results show the improvement of the proposed approach over the existing methods. [ABSTRACT FROM AUTHOR]

Terminology documents the knowledge of various subject fields. The paper describes the structure of knowledge and models the conceptual systems of ordinal numbers in English and Slovak language. In the theory of terminology, the designation can be expressed by a term, a symbol, and an appellation. Worth noticing is the fact that in the case of mathematical sciences, in algebra, both two types of designations are used, i.e., by a term and predominantly by a symbol. Mathematical science is precise and systemic. That is the reason why designation by the symbol is largely used in practice. An additional asset is that this type of designation is understandable across almost all languages and has an international character. It avoids ambiguation and misunderstandings. Moreover, the symbol saves time and is more economical. The paper focuses on bilingual conceptual systems of ordinal numbers. Conceptual systems facilitate understanding of relationships between the concepts, help to write definitions, and facilitate comparative analysis. Subject to bilingual conceptual and terminological analysis is also the decimal positional numeral system, the decimal referring to the use of 10 symbols 0,1,2,3,4,5,6,7,8,9 to construct all numbers. The comparative analysis contrasts conceptual systems of ones, tens, hundreds, thousands, ten thousands, hundred thousands, and millions in the English and Slovak language. The bilingual conceptual system can be represented by symbols, as well as by the terms in both languages. The conceptual system represents the hierarchy between the concepts and superordinate and subordinate relationships. Such hierarchical attitude is described in terminology as a generic and partitive relationship. In terminology research, the following methods have been used: observation, excerption, conceptual analysis, term analysis, comparative analysis of bilingual conceptual systems, classification of ordinal numbers, and synthesis of data. As a result of contrastive terminology work, (excerption, harmonisation, and terminography), English and Slovak bilingual conceptual systems of ordinal numbers have been modelled and displayed in the model of formal and graphical representation of bilingual conceptual systems of ordinal numbers. Systems have shown similarities and differences between the mathematical concepts of both languages. In the case of terminography, the bilingual conceptual systems and terminology records have been ready for multiple uses, for example, glossary, dictionary, and database …). The English language belongs to analytical languages, the Slovak language to synthetical languages. Due to different language types, differences in the concepts are present. In the conclusion of contrastive terminological research in the conceptual systems, there are some findings and recommendations for practice. Findings help to model concept structures based on specialised knowledge of the field and clarify the relations between concepts. The conceptual system facilitates the comparative analysis of concepts and designations across languages and helps specialists in writing the definitions. [ABSTRACT FROM AUTHOR]

Completeness is one of the basic assumptions about the rational preference relation in classical decision theory. Strongly and weakly consistent preferences are presented by abandoning the completeness of the rational preference relation. Some expansion and contraction conditions are proposed and the relationships between these conditions of rationality are discussed. The relationships between the conditions of rationality and boundedly rational choice behavior based on strongly and weakly consistent preferences are analyzed and discussed. Furthermore, an example about the choices of chocolates with interval ordinal numbers is given to explain some of the main conclusions in this paper. The results can be used as references for the study of boundedly rational decisions. [ABSTRACT FROM AUTHOR]

Sections 1 through 4 define, in the usual inductive style, various classes of object including one which is called the "combinatory terms of polymorphic type". Section 5 defines a reduction relation on these terms. Section 6 shows that the weak normalizability of the combinatory terms of polymorphic type entails the weak normalizability of the lambda terms of polymorphic type. The entailment is not vacuous, because the combinatory terms of polymorphic type are indeed weakly normalizable, as is proven in Sect. 7 using Tait and Girard's computability predicates. The remainder of the paper is devoted to arguing that interesting consequences would follow from the existence of an "ordinally informative" proof, i.e. one which uses transfinite induction over recursive ordinal numbers and otherwise finitary methods, of normalizability for as large a class of the terms as possible. [ABSTRACT FROM AUTHOR]

Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck's definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice "Univers" (par N. Bourbaki) de l'Exposé I. "PREFAISCE-AUX") [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML's definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of "trivial" and "non-trivial" Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. The following result links the universes U0 (FinSETS) and U1 (SETS): Grothendieck Universe ω = Grothendieck Universe U 0 = U 1 {\rm{Grothendieck}}\,{\rm{Universe}}\,\omega = {\rm{Grothendieck}}\,{\rm{Universe}}\,{{\bf{U}}_0} = {{\bf{U}}_1} Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, ncatlab.org [24]). Grothendieck's universe is referenced in current works: "Assuming the existence of a sufficient supply of (Grothendieck) univers", Jacob Lurie in "Higher Topos Theory" [17], "Annexe B – Some results on Grothendieck universes", Olivia Caramello and Riccardo Zanfa in "Relative topos theory via stacks" [13], "Remark 1.1.5 (quoting Michael Shulman [30])", Emily Riehl in "Category theory in Context" [29], and more specifically "Strict Universes for Grothendieck Topoi" [14]. [ABSTRACT FROM AUTHOR]

Background: To provide empirically based guidance for substituting partner number categories in large MSM surveys with mean numbers of sexual and condomless anal intercourse (CAI) partners in a secondary analysis of survey data.Methods: We collated data on numbers of sexual and CAI partners reported in a continuous scale (write-in number) in thirteen MSM surveys on sexual health and behaviour across 17 countries. Pooled descriptive statistics for the number of sexual and CAI partners during the last twelve (N = 55,180) and 6 months (N = 31,759) were calculated for two sets of categories commonly used in reporting numbers of sexual partners in sexual behaviour surveys.Results: The pooled mean number of partners in the previous 12 months for the total sample was 15.8 partners (SD = 36.6), while the median number of partners was 5 (IQR = 2-15). Means for number of partners in the previous 12 months for the first set of categories were: 16.4 for 11-20 partners (SD = 3.3); 27.8 for 21-30 (SD = 2.8); 38.6 for 31-40 (SD = 2.4); 49.6 for 41-50 (SD = 1.5); and 128.2 for 'more than 50' (SD = 98.1). Alternative upper cut-offs: 43.4 for 'more than 10' (SD = 57.7); 65.3 for 'more than 20' (SD = 70.3). Self-reported partner numbers for both time frames consistently exceeded 200 or 300. While there was substantial variation of overall means across surveys, the means for all chosen categories were very similar. Partner numbers above nine mainly clustered at multiples of tens, regardless of the selected time frame. The overall means for CAI partners were lower than those for sexual partners; however, such difference was completely absent from all categories beyond ten sexual and CAI partners.Conclusions: Clustering of reported partner numbers confirm common MSM sexual behaviour surveys' questionnaire piloting feedback indicating that responses to numbers of sexual partners beyond 10 are best guesses rather than precise counts, but large partner numbers above typical upper cut-offs are common. [ABSTRACT FROM AUTHOR]

Recent years have witnessed an increase in research on how numeral ordering skills relate to children's and adults' mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers. [ABSTRACT FROM AUTHOR]

A nonequilibrium Auger transition consisting of a vacancy in the valence layer being filled within ~10–14 s by a probe electron that receives the energy released by the transition of a valence electron to the K-shell and emits it from the surface is investigated. The nonequilibrium Auger spectrum from the surface of thin (8–10 atomic layers) films of silver oxide (Ag2O) on Si3N4 substrate at probe-radiation energy 8–10 keV is obtained for the first time. The transition probability of the escape of nonequilibrium Auger electrons is analyzed as a function of the ordinal atomic number. The amplitude peaks characterizing the contribution of nonequilibrium Auger electrons to the recorded signal are established. The probability, relaxation time, and capture cross section of a free charge carrier by a neutral vacancy are calculated based on a mathematical model describing the nonequilibrium Auger-process dynamics. [ABSTRACT FROM AUTHOR]

What are the cognitive mechanisms supporting non-symbolic and symbolic order processing? Preliminary evidence suggests that non-symbolic and symbolic order processing are partly distinct constructs. The precise mechanisms supporting these skills, however, are still unclear. Moreover, predictive patterns may undergo dynamic developmental changes during the first years of formal schooling. This study investigates the contribution of theoretically relevant constructs (non-symbolic and symbolic magnitude comparison, counting and storage and manipulation components of verbal and visuo-spatial working memory) to performance and developmental change in non-symbolic and symbolic numerical order processing. We followed 157 children longitudinally from Grade 1 to 3. In the order judgement tasks, children decided whether or not triplets of dots or digits were arranged in numerically ascending order. Non-symbolic magnitude comparison and visuo-spatial manipulation were significant predictors of initial performance in both non-symbolic and symbolic ordering. In line with our expectations, counting skills contributed additional variance to the prediction of symbolic, but not of non-symbolic ordering. Developmental change in ordering performance from Grade 1 to 2 was predicted by symbolic comparison skills and visuo-spatial manipulation. None of the predictors explained variance in developmental change from Grade 2 to 3. Taken together, the present results provide robust evidence for a general involvement of pair-wise magnitude comparison and visuo-spatial manipulation in numerical ordering, irrespective of the number format. Importantly, counting-based mechanisms appear to be a unique predictor of symbolic ordering. We thus conclude that there is only a partial overlap of the cognitive mechanisms underlying non-symbolic and symbolic order processing. [ABSTRACT FROM AUTHOR]

Non-symbolic numerical abilities are widespread among vertebrates due to their important adaptive value. Moreover, these abilities were considered peculiar of vertebrate species as numerical competence is regarded as cognitively sophisticated. However, recent evidence convincingly showed that this is not the case: invertebrates, with their limited number of neurons, proved able to successfully discriminate different quantities (e.g., of prey), to use the ordinal property of numbers, to solve arithmetic operations as addition and subtraction and even to master the concept of zero numerosity. To date, though, the debate is still open on the presence and the nature of a «sense of number» in invertebrates. Whether this is peculiar for discrete countable quantities (numerosities) or whether this is part of a more general magnitude system dealing with both discrete and continuous quantities, as hypothesized for humans and other vertebrates. Here we reviewed the main studies on numerical abilities of invertebrates, discussing in particular the recent findings supporting the hypothesis of a general mechanism that allows for processing of both discrete (i.e., number) and continuous dimensions (e.g., space). • Non-symbolic numerical abilities are widespread among vertebrate and invertebrate species. • Invertebrates are able to discriminate numerically different arrays, perform simple arithmetic operation and master the concept of zero numerosity. • Previous findings support the existence of a magnitude system in humans and other animals that allows the processing of both discrete (i.e., number) and continuous dimensions (e.g., space). • The hypothesis of the presence of a general sense of magnitude has been suggested for invertebrate species as well. [ABSTRACT FROM AUTHOR]